THE UNLIKELY UNION OF PARTITIONS AND DIVISORS Abdulkadir Hasse, Thomas J. Osler, Mathematics Departmet ad Tirupathi R. Chadrupatla, Mechaical Egieerig Rowa Uiversity Glassboro, NJ 828 I the multiplicative umber theory we decompose a atural umber ito prime factors = p p 2 p 3... p k ad cosider the cosequeces. I the additive theory we decompose a atural umber ito a sum of elemets from some set. For eample we could try to epress as a sum of squares. I [3], ad [4], the authors treated the properties of the partitio fuctio, which is a good eample of additive umber theory. A good eample of fuctios studied i multiplicative theory is the divisor fuctio σ ( ). σ ( ) is defied as the sum of the positive divisors of. For eample the divisors of 6 are, 2, 3, ad 6. Thus the sum of the divisors of 6 is σ ( 6) = + 2+ 3+ 6= 2. Now divisors of umbers are related to primes, ad primes seem urelated to partitios. We are ot surprised that partitios satisfy a recursio relatio. We do ot epect σ ( ) to satisfy a recursio relatio. What do the divisors of have to do with the divisors of, 2,...? Yet Euler showed that σ ( ) satisfies the same recursio relatio as does p( ), the partitio of.(see []) Oly σ ( ) is differet from p(). Euler was astoished at this result, ad you ca read a traslatio of his ow words i Polya [5] ad i Youg [8]. There are eve relatios marryig the two fuctios such as (Schroeder [6]) p ( ) = σ ( k) p ( k). k = I this paper we will eamie this ad other properties of σ ( ) ad p().
. BASIC PROPERTIES OF THE DIVISOR FUNCTION We use the covetio that σ () =. It is clear that σ ( p) = p + for ay prime umber p, sice the oly positive divisors of p are ad p. Also the oly divisors of 2 p are, p ad 2 p. 2 2 Thus σ ( p ) = p + p+ ad little bit of algebra shows that this ca be epressed as 2 2 σ ( p ) = p + p+ = 3 p. It should ow easy to prove p Lemma. If p is a prime ad k is a oegative iteger, the p k k k p + σ ( p ) = p + p+ = The followig is a stadard theorem i Number Theory tetbooks. (See [9]). Lemma 2. σ ( ) is multiplicative, that is, if m ad are relatively prime( they have o commo divisor other tha ), the σ ( m) = σ( ) σ( m). A immediate cosequece of Lemma ad 2 is the followig formula that ca be used to evaluate the sum of the divisors of a give iteger. k k2 k m Propositio. If = p p2 p m, where p, p2,, p m are primes ad k, k2,, k m are oegative itegers, the k+ k2+ km + k k2 k p p2 p m m 2 m p p2 pm σ( ) = σ( p p p ) = Propositio has a misleadig simplicity. Suppose we take 23 = 2 +. What are the prime factors of? I other words, if is a large umber whose prime factors caot easily be obtaied, the formula becomes less useful. It is i this sese that we ow look at the recurrece relatios that σ ( ) satisfies.
2. RECURENCE RELATIONS INVOLVING THE DIVISOR FUNCTION Let p( ) be the partitio of, that is, the umber of ways we ca write as a sum of positive itegers. We defie p ( ) = if. Let k(3k ) f( k) =. Euler proved the 2 followig recurrece relatio for p( ). { )} k + (.) ( ) ( p( ) = ( ) p f( k) + p f( k) k = The mai tool that Euler used to prove this formula was the fuctio give by the ifiite product properties: g ( ) = ( ). He showed that this fuctio has the followig two remarkable = ( ) ( ) (A) ( ) ( f k f k g= ) = ( ) (B) = = = g ( ) k = = ( ) = p( ) For more o this, see [3]. We ow defie the so-called Lambert series ad look at the relatio of this series with partitio ad divisor fuctios. Let { a ( )} be a sequece of real umber. The Lambert s series associated with{ a ( )} is defied by La ( ) = a( ) = This Lambert series is Taylor series epasio give by where L ( ) = A( ), a = A ( ) = ad ( ) (the summatio is take over the positive divisors of.) Note the that if A ( ) = σ ( ). Thus we have the followig result. d a ( ) = the
Lemma 3. = σ ( ) = = For more iterestig properties ad the may other applicatios of Lambert series the reader is hereby ivited to idulge i the classic book of Kopp [2]. We are ow i a positio to prove our first recurrece relatio that coects σ ( ) ad p( ). Let us rewrite the ifiite ( ) f ( k) f ( k) product of Euler as g ( ) = = ( ) = e. Theorem. (.2) = σ ( ) = ke p( k) k= = The sum of divisor ad partitio fuctio are related by the followig formula k = k Proof: If we take the logarithmic differetiatio of g ( ) =, ad multiply the resultig equatio by, we get (.3) O the other had, g ( ) = e = g'( ) = g ( ) implies that = = ( = g'( ) = e. We ow recall that k = p( ). Usig Lemma 3 ad substitutig these last two series for g ( ) = g '( ) ad i (.3), we get g ( ) (.4) k p( ) e = = σ ( ) = = = = Multiplyig out the two series o the left side of (.4)ad comparig coefficiets proves the theorem. I the same lie of reasoig we ca show the followig recurrece. Theorem 2. The divisor fuctio satisfies the recurrece relatio (.5) σ( ) = e e kσ ( k) Proof: We have oted that = k = = g '( ) = σ ( ). Multiply both sides by g() to get g ( ) g'( ) = g( ) σ ( ). Now use the fact that g ( ) = e ad g'( ) = e to get e = e σ ( ). Sice e =, the theorem follows from comparig = = = coefficiets of the two power series. = ) =
Theorem 3. The partitio fuctio ad the divisor fuctio are also related by the formula (.6) Proof: (.7) p( ) = σ ( k) p( k) k = Defie F( ) = = = p( ). The g ( ) = ( g ( )) 2 ( ) = g'( ) g'( ) g'( ) F'( ) = = F( ) g ( ) g ( ) = g ( ) O the other had, we have F'( ) = p( ) = g'( ) g ( ) = σ. Also F( ) p( ) implies that = ( ). Substitutig all these i (.7), we get (.8) ( ) p = σ ( ) p( ) = = = Multiply (.8) by to get ( ) p = σ ( ) p( ) = =. = Multiplyig the ifiite series o the right ad compare coefficiet to get the formula i the theorem. Remark: [3] cotais a combiatorial proof of Theorem 3. He first shows by a combiatorial argumet that m m= k= = p( ) = m p( km). Covertig the double sum ito a sigle sum the gives the formula i Theorem 3. 3. QBASIC PROGRAM FOR THE SUM OF DIVISORS Formula (.) ca be rewritte as p ( ) = p ( ) + p ( 2) p ( 5) p ( 7) p ( 2) + p ( 5) p ( 22) p ( 26) + = The authors used this recurrece formula i [3] to write a simple QBASIC program t costruct a table for the values of p(). We ca use the same program with a slight modificatio to geerate a table of values for σ ( ). Here is the modified program of [Osler ] that uses formula (.2). Program : Calculate Partitios ad Sum of Divisor
'Calculate partitios of N, P()=p(N), ad the sum of divisors DN ( ) = σ ( N) 'eactly up to P(3) ad D(3). 'Set double precisio, dimesio array P ad D, iitialize P ad D 'Mai loop, for each N fid P(N) ad D(N). 9 CLS DEFDBL A-Z DIM P(4) DIM D(4) 2 P() = 2 D() = INPUT "ENTER A POSITIVE INTEGER", J 'Mai loop, for each N fid P(N)ad D(N) 2 FOR N = TO J 2 SIGN = 25 P(N) = 26 D(N) = 22 FOR K = TO N 'Calculate two terms i recursio relatio for P(N)ad D(N) 23 F = K * (3 * K - ) / 2 'Eit loop if argumet egative 24 IF N - F < THEN GOTO 4 25 P(N) = P(N) + SIGN * P(N - F) 25 D(N) = D(N) + F * SIGN * P(N - F) 26 F = K * (3 * K + ) / 2 'Eit loop if argumet egative 27 IF N - F < THEN GOTO 4 28 P(N) = P(N) + SIGN * P(N - F) 28 D(N) = D(N) + F * SIGN * P(N - F) 29 SIGN = -SIGN 3 NEXT K 'Prit results 4 PRINT N, P(N), D(N) 'Pause after pritig 2 lies o the scree 45 IF 2 * INT(N / 2) = N THEN INPUT A$: CLS 5 NEXT N REFERENCES
. Grosswald, Emil, Topics from the Theory of Numbers, MacMilla Co., N. Y., 966, p. 223. 2. Kopp, K. The Theory ad Applicatios of Ifiite Series. 3. Hasse, Abdul ad Osler, Thomas, Playig With Partitios O The Computer Mathematics ad Computer Educatio. Vol. 35 No., page 5-7 (Witer 2) 4. Chadrupatla, T R; Hasse, Abdul; ad Osler, Thomas, A Table of Partitio Fuctio. Mathematical Spectrum, Vol. 34 No 3, page 55-57 (2/22) 5. Polya, G., Mathematics ad Plausible Reasoig, Vol., Iductio ad Aalogy i Mathematics, Priceto Uiversity Press, Priceto, NJ, 954, pp. 9-7 6. Schroeder, M. H., Number Theory i Sciece ad Commuicatio, (Secod Ed.), Spriger-Verlag, New York, 986, p.23. 7. Specer, Doald D., Eplorig Number Theory with Microcomputers, Camelot Publishig, Orlado Beach, Florida, 989. 8. Youg, Robert M., Ecursios i Calculus, A Iterplay of the Cotiuous ad the Discrete, Mathematical Associatio of America, 992, pp. 357-368. 9. Rose, Keeth, Elemetary Number Theory ad Its Applicatios, 4th ed., Addiso- Wesley Pub. Co.